TITLE: What are these two topologies?
QUESTION [6 upvotes]: I'm a beginner in Topology. Today, this came up my mind:
(1) For a set $X$, choose a subset $A\subseteq X$. Let $S\subseteq X$ be a closed set if and only if $(A\subseteq S)\vee (S=\emptyset)$. This is a topology on $X$.
(2) For a set $X$, let $S\subseteq X$ be a closed set if and only if $(S $ is finite$)\vee (S = X)$. This is another topology on $X$.
The questions are:
[1] What are these two topologies called?
[2] Do they have significance? Are they used somewhere?
First question on stackexchange. Thanks!

REPLY [5 votes]: Your second topology is known as the cofinite topology on $X$; it’s important because a topology $\tau$ on $X$ is $T_1$ if and only if $\tau$ contains the cofinite topology. As Ilya notes in the comments, it is also useful in the construction of counterexamples, at least when the examples are not required to be Hausdorff.
I don’t know of any name for your first topology. It’s the discrete topology on $X\setminus A$ together with the indigestible lump $A$, a set that behaves in most respects like a single point. The quotient space $X/A$ is simply a discrete space with one point corresponding to $A$ and one point for each point of $X\setminus A$.